It is, in many respects, less difficult/more secure than riding astride. There are several people here who do it. There was a thread about it a few months ago, I believe in the the "horse talk" section.
I haven't personally seen side saddle riding, but I have seen a show being streamed online that did have a side saddle class so it does still exist. I cannot remember if it was an Arabian or Saddlebred show.
I got to ride side saddle at the dude ranch I went to in Montana. One of their daughters had one and was letting the guests take a stab at it if they wanted. It was surprisingly comfy, but weird since I'm still kinda a beginner so I wasn't proficient at using the crop on the right side where my leg would normally be. So I was content to be led around, but it was still fun!